This book, written for undergraduate math students,
describes a striking connection between topology and
algebra, expressed by the theorem that 2D topological quantum field
theories are the same as commutative Frobenius algebras. The precise
formulation of the theorem and its proof is given in terms of monoidal
categories, and the main purpose of the book is to develop these
concepts from an elementary level, and more generally serve as an
introduction to categorical viewpoints in mathematics. Rather than
just proving the theorem, it is shown how the result fits into a more
general pattern concerning universal monoidal categories for algebraic
structures. Throughout, the emphasis is on the interplay between
algebra and topology, with graphical interpretation of algebraic
operations, and topological structures described
algebraically in terms of generators and relations. The picture on
the cover is the topological expression of the main axiom for a
Frobenius algebra.
The book was reviewed in the Newsletter of the European Mathematical
Society, March 2005. The reviewer (who signs himself mm) writes:

The book is very well written and organized. I warmly recommend it as an
introduction to basic techniques of algebraic geometry.

This is a fantastic and highly surprising recommendation, since the book is
not at all about algebraic geometry!

(A more serious review, by David Yetter, appeared in the Bulletin of the
London Mathematical Society, 36 (2004).)

Page 75--77, 1.4.39: FLAW IN PROOF -- thanks to Chris Heunen and
Jamie Vicary for pointing this out. There is a confusion between
rectangular regions and connected components: the latter can be trapped
inside each other without forming rectangular regions. A new proof
strategy seems to be required. A correct proof (I believe) is given
here.

Page 33--34, Exercise 7-9: MATHEMATICAL ERROR -- thanks a lot to Jim Bryan for
pointing it out. These three exercises are meaningless since the TQFT in
question cannot exist. It corresponds to a non-commutative Frobenius
algebra.

Page 42, Proposition 1.3.23: MATHEMATICAL ERROR -- Thanks a lot to
Chris Schommer-Pries
and Gard Spreemann (independently) for pointing this out and correcting the
error. The correct statement (Milnor, "Lectures on the h-Cobordism Theorem",
Thm.1.9) is that two diffeomorphisms induce the same
cobordism class iff they are pseudo-isotopic. (The notion of pseudo-isotopy
is actually what was used in the proof of 1.3.23.)

Here is the replacement (for the proposition and its proof, as well as the
paragraphs immediately before and after):

Two questions are natural at this point. Does every invertible
cobordism arise from a diffeomorphism? When do two diffeomorphisms
give the same cobordism class? The first question we will answer
(affirmatively) only in the $2$-dimensional case, in the next section.
The second question is settled by the next proposition.
It relies on the notion of pseudo-isotopy: two diffeomorphisms
$\psi_0 : \Sigma_0 \to \Sigma_1$ and $\psi_1 : \Sigma_0 \to \Sigma_1$
are said to be {\em pseudo-isotopic} if there is a diffeomorphism
$\Psi: \Sigma_0 \times I \to \Sigma_1 \times I$ which agrees with $\psi_0$
in one end of the cylinder and with $\psi_1$ in the other:
\begin{diagram}[w=6ex,h=3ex,tight]
&& \Sigma_1\times I && \\
& \ruTo^{(\psi_0,0)} && \luTo^{(\psi_1,1)} & \\
\Sigma_0 && \uTo>{\Psi} && \Sigma_0 \\
& \rdTo_{(\id,0)} && \ldTo_{(\id,1)} & \\
&& \Sigma_0\times I &&&&
\end{diagram}
\begin{prop}
Two diffeomorphisms $\Sigma_0 \topile \Sigma_1$ induce the same cobordism
class $\Sigma_0 \cobto \Sigma_1$ if and only if they are pseudo-isotopic.
\end{prop}
\begin{dem}
Suppose $\psi_0, \psi_1 : \Sigma_0 \topile \Sigma_1$ are pseudo-isotopic.
Compose the above diagram with
\begin{diagram}[w=6ex,h=4.5ex,tight]
\Sigma_0 &\lTo^{\psi_1^{-1}} & \Sigma_1
\end{diagram}
on the right, getting
\begin{diagram}[w=6ex,h=3ex,tight]
&& \Sigma_1\times I && \\
& \ruTo^{(\psi_0,0)} && \luTo^{(\id,1)} & \\
\Sigma_0 && \uTo>{\Psi} && \Sigma_1 \\
& \rdTo_{(\id,0)} && \ldTo_{(\psi_1^{-1},1)} & \\
&& \Sigma_0\times I &&&&
\end{diagram}
The upper part of this diagram is the cobordism class induced by $\psi_0$;
the lower part is the cobordism induced by $\psi_1$ (in the `backward'
convention), and $\Psi$ expresses that they are equivalent. The converse
implication amounts to reversing the argument.
\end{dem}
So in particular, a cobordism $\Sigma\cobto \Sigma$ induced by a
diffeomorphism $\psi:\Sigma \isopil \Sigma$ is the identity if and only if
$\psi$ is pseudo-isotopic to the identity. As an example of a diffeomorphism
which is not pseudo-isotopic to the identity, take the twist diffeomorphism
$\Sigma\disju \Sigma \to \Sigma\disju \Sigma$ which interchanges the two
copies of $\Sigma$.

I should also take the opportunity to eliminate the abusive notation
introduced in 1.3.22, and write $(\phi,0)$ instead of just $\phi$ in the
diagram on page 46, etc, like in the above replacement.

Page 51, line 26: 'interchanging label' should be 'interchanging labels'.

Page 59, four lines from the bottom: there should be a blank line
before the line that starts 'So instead...'.

Page 68, line 4: the reference to [Hirsch] should be to 9.3.4, not to
4.4.2. Thanks to Marius Thaule for pointing it out.

Page 70, last line:, the reference to 1.4.30 should rightly be to
1.4.32.

Page 74, 1.4.37: The parenthesis
"(Note that since the surface is connected, in fact every occurrence
of [cap] must be to the left of a [mult]...)"
should simply be deleted, as it is completely superfluous and actually
wrong: it is possible to have a single [cap] and no [mult].

Page 101: In the displayed formula in 2.2.19, there are two strange
quote characters. They should not be there at all.

Page 111: in the diamond shaped diagram, the two maps labelled 'mu'
should rather be labelled 'beta'.

Page 122, l.-7: The proof of Prop. 2.3.29 promises first to show that
a certain map has 'epsilon' as counit, and satisfies the Frobenius
equation. But the first promise is not fulfilled (thanks Carlos Moraga
for pointing this out). There should be a sentence saying

That $\epsilon$ is a counit for $\delta \sigma$ follows by first using
naturality of
$\sigma$ with respect to $\epsilon$, and then the fact that $\epsilon$ is
a counit for $\delta$.

Page 130, line 1: it should be y^2+1 instead of y^2-1. (Thanks Carlos
again.)

Page 136, line 8 (first diagram): The upper right-hand 'A' should
be 'A \tensor A'. (This typo illustrates one advantage of graphical
notation over symbolic: this sort of 'syntax error' could not possibly
have occurred in graphical notation, like in the drawing just to the
right of the diagram.)

Page 153, last diagram in 3.2.10: in the right-hand side of the
diagram, the vertical map should be from V'^n to V', and its name should
be \mu'^(n).

Page 155, line 7: It says: 'the result is not the same'. It should
be 'the result is the same'.

Page 163, line 13: 'that that' should be 'that'.

Page 166, in 3.2.46: the description of the morphisms in the category
of graded vector spaces is misleading since it suggests that only
maps of degree 0 are allowed. In fact homogeneous maps of any degree
should be allowed. This is the notion used elsewhere in the book,
in 2.2.23 and in 3.3.3.

Page 169, second line of Exercise 8: It says 'a family of maps'.
It should be 'a family of invertible maps'.

Page 177, Exercise 7 and Exercise 8: MATHEMATICAL ERROR
-- thanks a lot to Jim Bryan for pointing it out.
The statement of the exercise 7 is false, and exercise 8 is therefore
meaningless. I propose the following replacement exercises:

7. Consider now the $1$-dimensional Frobenius algebra $k$ over $k$, with
Frobenius form $1 \mapsto u$. Show that the corresponding TQFT associates
the invariant $u^{1-g}$ to a closed surface of genus $g$. Make sense of the
formula also for nonconnected surfaces (hint: use 1.4.14).

8. Generalise the preceding exercise as follows:
Let $(A,\epsilon)$ be a commutative Frobenius algebra, and let $a_g$
denote the invariant associated to a closed genus-$g$ surface by the
corresponding TQFT. Show that if the Frobenius form $\epsilon$
is adjusted by a unit factor $u$ (cf.~Lemma 2.2.8),
then the corresponding invariants $a_g$ will be adjusted by a
factor $u^{1-g}$.

Page 186, line 8: 'composition of face maps' should be replaced by
'composition of degeneracy maps'. (Thanks Carlos.)

Page 235, in bib item [32], Lawvere's paper on ordinal sums, the
publication year should be 1969, not 1967. (1967 is the year of the
seminar from where the publication orginates.)

I think it should be mentioned as Proposition 1.4.42 that
assuming commutativity and cocommutativity as well as the naturality
of the twist, then the two Frobenius equations imply each other. An
extra exercise 6 (also in Section 1.4) should then be added asking to
supply the proof, an instructive routine calculation.

is the modern categorical
characterisation of what it means to be a Frobenius algebra (Chapter 2), a
characterisation that makes sense in any monoidal category, and hence more
generally defines a notion of Frobenius object in any monoidal category.
The category of 2-dimensional cobordisms (Chapter 1) is the free symmetric
monoidal category on a commutative Frobenius object (Chapter 3). The
classical characterisation of Frobenius algebra uses concepts like kernel
and ideal that do not make sense outside a narrow abelian setting.

Bill Lawvere knew about the categorical characterisation of Frobenius
algebras in 1967, but he did not explicitly write the Frobenius equation.
In Chapter 2, I write that the first explicit appearance of the Frobenius
equation is in the lecture notes of Quinn (published in 1995, lectures
from 1991). This turns out to be wrong:

Aurelio Carboni and Bob Walters have pointed out to me (March 2006) that
the first explicit appearance of the equation is in A. Carboni, R.F.C.
Walters, Cartesian bicategories I, J. Pure Appl. Alg. 49
(1987), 11-32 (submitted February 1985). That paper studied the equation
from another viewpoint (categories of relations), but without realising
that it is also the equation characterising Frobenius algebras (in
particular the authors were unaware of Lawvere's remark at the time). The
pieces came together shortly after, according to the following historical
account of the equation, which is very interesting and lively, and, it
seems to me, very illustrative for the category theory community. I am
grateful for their permission to reproduce it here. The text is also
available from Bob Walter's
Blog.

Date: Thu, 09 Mar 2006 14:01:25 +0100
From: RFC Walters
Subject: Some categorical history of the Frobenius equation
To: Joachim Kock
Cc: Aurelio Carboni
Dear Joachim,
We have just been reading your very pleasant book about the relation
between Frobenius algebras and cobordism. Perhaps you may be interested
in some further history, from the categorical community, of the
Frobenius equation, arising from a different line of research, and
curiously not mentioned in the article by Ross Street, "An Australian
conspectus of higher categories, Institute for Mathematics and
Applications Summer Program, n-categories: Foundations and Applications,
June, 2004".
One of us (Bob Walters) has written a blog entry (at
http://rfcwalters.blogspot.com)
recounting the story as we know it. We include that below.
As far as we know we were the first to explicitly publish the equation
in 1987 (submitted February 1985), not Quinn as you report. But of
course there may be even earlier occurrences, and there is the
equivalent set of equations published by Lawvere in 1969.
The other fact is that Joyal certainly knew the connection with
cobordism when we talked with him in Louvain-la-Neuve in 1987.
best regards,
Aurelio Carboni and Robert FC Walters
Como, 9 March 2006
----------------------------------------------------------------------
>From a posting in blog http://rfcwalters.blogspot.com Wednesday,
February 15, 2006
History of an equation - (1 tensor delta)(nabla tensor 1)=(nabla)(delta)
This is a personal history of the equation
(1 tensor delta)(nabla tensor 1)=(nabla)(delta)
now called the Frobenius equation, or by computer scientists S=X.
1983 Milano:
Worked with Aurelio Carboni in Milano, and later in Sydney, on
characterizing the category of relations.
1985 Sydney:
We submitted to JPAA on 12th February the paper eventually published as
A. Carboni, R.F.C. Walters, Cartesian bicategories I, Journal of Pure
and Applied Algebra 49 (1987), pp. 11-32.
The main equation was the Frobenius law, called by us discreteness or
(D)(page 15).
1985 Isle of Thorns, Sussex:
Lectured on work with Carboni concentrating on importance of this new
equation - replacing Freyd's "modular law" (see Freyd' book "Categories,
Allegories"). Present in the audience were Joyal, Anders Kock, Lawvere,
Mac Lane, Pitts, Scedrov, Street. I asked the audience to state the
modular law, Joyal responded with the classical modular law, Pitts
finally wrote the law on the board, but mistakenly. Scedrov said "So
what?" to the new equation and "After all, the new law is equivalent to
the modular law". Nobody ventured to have seen the equation before.
(I asked Freyd in Gummersbach in 1981 where he had found the modular
law, and he replied that he found it by looking at all the small laws on
relations involving intersection, composition and opposite, until he
found the shortest one that generated the rest. We believe that this
law actually occurs also in Tarski,
A. Tarski, On the Calculus of Relations, J. of Symbolic Logic 6(3), pp.
73-89 (1941),
but certainly in the book "Set theory without variables" by Tarski and
Givant, though not in the central role that Freyd emphasised.)
At this Sussex meeting Ross Street reported on his discovery with
Andre Joyal of braided monoidal categories (in the birth of which we
also played a part - lecture by RFC Walters, Sydney Category Seminar,
On a conversation with Aurelio Carboni and Bill Lawvere: the
Eckmann-Hilton argument one-dimension up, 26th January 1983). This
disovery was a major impulse towards the study of geometry and higher
dimensional categories.
1987 Louvain-la-Neuve Conference:
I lectured on well-supported compact closed categories - every object
has a structure satisfying the equation S=X, plus diamond=1. Aurelio
spoke about his discovery that adding the axiom diamond=1 to the
commutative and Frobenius equations characterizes commutative separable
algebras, later reported in
A. Carboni, Matrices, relations, and group representations, J. Alg. Vol
136, No 2,1991 (submitted in 1988)
(see in particular, the theorem and the remark in section 2).
After Aurelio's lecture Andre Joyal stood up and declared that "These
equations will never be forgotten!".
At this, Sammy Eilenberg rather ostentatiously rose and left the lecture
- perhaps the equation occurs already in Cartan-Eilenberg?
Andre pointed out to us the geometry of the equation - drawing lots of
2-cobordisms.
During the conference in a discussion in a bar with Joyal, Bill Lawvere
and others, Bill recalled that he had written equations for Frobenius
algebras in his work
F.W. Lawvere, Ordinal Sums and Equational Doctrines, Springer Lecture
Notes in Mathematics No. 80, Springer-Verlag (1969), 141-155.
The equations did not incude S=X, diamond=1, or symmetry, but the
equation S=X is easily deducible (see Carboni, "Matrices...", section
2). Bill's interest, as ours, was to discover a general notion of
self-dual object. In Freyd's work there is instead the assumption of an
involution satisfying X^opp=X.

The paragraph mentioning braided monoidal categories was not in the
original email. It was added by Bob Walters a few days later. Bob also
observes that Joyal's cobordism interpretation of Frobenius by 1987 must be
seen in the context of the explosion of work on geometric interpretation of
categorical equations (tangles, ribbons etc) that ensued from the advent of
braided monoidal categories.